In this video, we’ll discuss Shkadov thrusters: a method of moving stars, star systems, and even entire galaxies.
The lack of oxygen in Mars' atmosphere and running liquid water on its surface is very inconveniant for any humans living their since oxygen and liquid water are necessary for humans to survive. Fortunatelly, there is an abundance of frozen water on Mars' surface. In this lesson, we'll discuss various techniques which can be used to extract all of this water. Once the water is obtained, by performing electrolysis on the water we can distill all of the oxygen from that water we need.
In this section, we'll find the solution (which is a curve x(y)) to the Euler-Lagrange equation where the quantity being minimized is the arc length between any two points on a cylinder. That might sound like a mouthful, but we'll see that hopefully it isn't too bad.
Guass's law: for any charge distribution, whether a system of a finite number of point charges or a continuous distribution of charges, if I draw any arbitrary closed surface (which can be of any shape, size, and location), then the only contributions to the electric flux through that closed surface will be due to only the charges \(q_{enc}\) enclosed within that surface and that electric flux is given by the amount \(\frac{q_{enc}}{\epsilon_0}\). It is important to appreciate the word any to fully grasp the applicability of Guass's law: it applies to any closed surface and to any distribution of charge.
Everything in this section will summarize some—a fairly good portion—of what we'll be studying in much more detail in the subsequent sections.
Column's law describes the electric force between any distribution and number of charged particles. It is derived empirically and falls off inversely with the square of the distance, analogous to Newton's law of gravity. The practical usefulness of this law is that it allows one to determine the effect that any distribution of charge has on any other distribution of charge; this is analogous to how Newton's law of gravity allows one to determine the effects of one mass distribution on another. One very gradually learns this by solving many problems. But in this section, we'll start off with Column's law and then proceed to do some algebra and calculus in order to derive equations which describe the electric force that systems and continuous charge distributions exert on an individual particle.
In this section, we'll derive the Euler-Lagrange equation. The Euler-Lagrange equation is a differential equation whose solution minimizes some quantity which is a functional. There are many applications of this equation (such as the two in the subsequent sections) but perhaps the most fruitful one was generalizing Newton's second law. This equation, together with Hamilton's principle, allows us to generalize Newtonian mechanics and find the motion of systems using generalized coordinates—all of a sudden, double-pendulums don't seem so scary.
In this lesson, we'll see that Guass' law makes the same prediction about the electric field produced by a point charge as Column's law.
Unlike the flux of the velocity field of a fluid, the notion of electric flux is much more abstract. Like concepts such as energy, it is very abstract but nonetheless very useful in analyzing certain situations. In this section, we derive the electric flux of any electric field through any arbitrary surface. Only later one, when we discuss Guass's Law, will we learn about the usefulness of this idea.
Noether's Theorem is one of the most profound theorems in all of science, let alone physics. It implies that the very fabric of space and time give rise to the laws of nature and lead the mathematician David Hilbert to independently discover the general theory of relativity, alongside Einstein. Noether's theorem, in simplest terms, is the statement that for every symmetry (what a symmetry is is something we'll get to in this lecture) there is an associated conservation law. Our goal in this lecture will be to prove that and show that various different conservation laws follow from an associated symmetry.
The difference between kinematics, dynamics, and mechanics and what each of these terms means is a common point of confusion. But in the most basic sense, the meanings and differences of each of these terms can be described as follows: kinematics describes the motion of an object without consideration of what caused that motion; dynamics describes what caused that motion (namely, a force) without consideration of what the motion of the object is; and mechanics combines the two and describes both the motion and cause of the motion.
Galileo once pondered how one could describe the motion of cannonballs and other projectiles moving near Earth's surface. Since Aristotle, philosophers had tried for millennia to tackle this problem but it wasn't until Newton invented calculus and formulated the laws of classical mechanics when this question was finally answered. In this lesson, we'll use calculus and Newton's laws to answer this nearly 2,500 year old problem and derive the equations of motion governing projectile motion.
Newton's law of gravity has been described as one of the greatest achievements in human thought of all time. It says that everything in the universe is, quite literally, connected. It must've been an astonishing realization to Newton that a grain of dust in his room exerts a slight "pull" on all of the stars and galaxies in the universe. In the words of Paul Dirac: "Pick a flower on Earth and you move the farthest star."
In this section, we delve into some very fundamental ideas which, although expressed with respect to different quantities, are the basis of not only classical mechanics but also relativistic and quantum mechanics as well. These three areas of physics describe the universe within different ranges of parameters: quantum mechanics deals with things on the smallest scale; classical mechanics deals mostly with the size-range of macroscopic objects that we are all familiar with; and general relativity describes the most massive objects in the universe. All of these areas of physics involve some notion of inertia, a state which does not change; each of these areas also have an equation of motion which describes how inertial motion changes. We'll discover that the notions of inertial motion and the change in inertial motion can be expressed in terms of a very small number of elementary rules; and yet, these rules encapsulate a myriad of predictions which encompasses all observable phenomena within a given range of parameters.
Kinematics is the study of the position (represented by the position vector \(\vec{R}(t)\)) of an object as a function of time. The position vector can be used to define other quantities such as velocity \(\vec{v}\) and acceleration \(\vec{a}\); all three of these quantities, together, can fully describe the motion of any object. In this lesson, we'll study these three fundamental quantities of kinematics.
In this lesson, we'll discuss the foundation of Classical mechanics: namely, Newton's three laws of motion. Everything else that we discuss in Classical mechanics will be based on these principles.
In this lesson, we'll show that the principle of momentum conservation can be derived using Newton's laws of motion.
In this lesson, we'll analyze the motion of object's falling near the Earth's surface at slow velocities.
Many of the concepts we use in physics are very abstract and "non-visualizable." But, nonetheless, they can be applied to tell us a great deal about how the universe works. For example, the concepts of angular momentum and the conservation of angular momentum are very abstract and it might, at least initially, not seem to have much to do with anything based in physical reality—as Feynman probably would've said, the latter is just a number that we keep measuring to be the same. But when these concepts are applied, they actually "say" or predict a lot about how the universe works: this law requires everything from solar systems being flat to a spinning ice skater rotating faster as they bring their arms in.
Scaling factor a(t) and Hubble's Parameter H(t)
This video was produced by David Butler. For a copy of the transcript of this video, visit: http://howfarawayisit.com/documents/
Shortly after the precise quantitative predictions of Einstein’s general relativity concerning the precession of Mercury’s perihelion and the deflection angle of rays of light passing the Sun, Einstein moved beyond investigations of the solar system and applied general relativity to the entire universe. He wondered what the effects of gravity would be due to all the masses and energy in the universe. This might seem like an impossible task, but Einstein greatly simplified matters by assuming that the distribution of all the matter in the universe was spatially uniform. He called this assumption the cosmological principle. This means that the distribution of all mass throughout space is homogenous and isotropic. If the mass distribution is homogenous, then if you draw a line in any direction which extends throughout all of space, all of the mass distribution along that line will be equally spaced; isotropy means that the distribution of mass is the same in all directions. If a distribution of mass is both homogenous and isotropic then it is equally spaced and the same in all direction. Later observations (in particular the Cosmic Microwave Background Radiation) proved that on the scale of hundreds of millions of light-years across space, the distribution of galaxies very nearly (up to very miniscule non-uniformities) is completely homogenous and isotropic; thus on this scale the cosmological principle is a reasonable idealization.
Figure 1: The coordinate value \(x^i\) assigned to each tick mark in the rectangular coordinate system above remains the same as the coordinate system stretches or contracts. Only the scaling factor \(a(t)\) changes when the coordinate system stretches or contracts.
Imagine that we draw a line through our galaxy that extends across space for hundreds of millions of light-years. Let’s label this line with equally spaced points which have fixed coordinate values \(x^1\). Imagine that embedded and attached to those points are point-masses (each having a mass \(m\)) which we can think of as galaxies. If we stretch or contract this line, the point-masses (galaxies) will either move away from or towards one another. The coordinate value \(x^1\) of each mass does not change since, as we stretch the line, the point embedded in the line and the galaxy remain “overlapping each other.” We shall, for simplicity, consider our galaxy to be located at the origin of the coordinate system at \(x^1 = 0\) although (as we will soon see) the choice of the origin is completely arbitrary. We define the distance between galaxies on this line to be \(D\equiv{a(t)∆x^1}\) where, based on this definition, the scaling factor \(a(t)\) is the distance \(D=a(t)\cdot1=a(t)\) between two galaxies separated by \(∆x^1=1\). (I repeat, the coordinate value \(x^1\) of each galaxies doesn’t change and, therefore, the “coordinate separation” \(∆x^1\) between galaxies doesn’t change.) We will assume that the masses along this line are homogeneously distributed which just means that all of the masses are, at all times \(t\), equally spaced. In other words, at all times \(t\), the distance \(D=a(t)(x^1 - x^1_0) = a(t)\) (where \(∆𝑥¹=(x^1 - x^1_0)=1\)) between any two galaxies on the line separated by \(∆x^1=1\) with any coordinates \(x^1\) and \(x^1_0\); this is just a mathematically precise way of saying that the distance \(D=a(t)\) between two galaxies separated by “one coordinate unit” doesn’t depend on where we are on the line (\(x^1\) and \(𝑥^1_0\) could be anything, the distance will still be the same.)
Let’s draw another line (at a right angle to the first) through our galaxy which, also, extends for hundreds of millions of light-years across space. Let’s also label this line with equally spaced points where galaxies of mass \(m\) sit on. We will also assume that the distribution of masses along this line is homogenous (meaning they are all equally spaced) and that the spacing between these points is the same as the spacing between the points on the other line (which means that the total mass distribution along both lines is isotropic). Isotropic just means that the distribution of mass is the same in all directions. The equation \(D=a(t)∆x^2\) is the distance \(D\) between two galaxies on the vertical line drawn in the picture. We can find the distance \(D\) between two galaxies with coordinates \((x^1_0, x^2_0)\) and \((x^1, x^2)\) using the Pythagorean Theorem. Their separation distance \(D_{x^1}\)along the horizontal line is \(D_{x^1}=a(t)∆x^1\) and their separation distance \(D_{x^2}\) along the vertical line is \(D_{x^2-axis}= a(t)∆x^2\). Using the Pythagorean Theorem, we see that \(D=\sqrt{(D_{x^1})+(D_{x^2})}\). To make this equation more compact, let \(∆r=\sqrt{(∆x^1)^2 +(∆x^2)^2}\) which we can think of as the “coordinate separation distance” which doesn’t change. Then we can write the distance as \(D=a(t)∆r\).
If we drew a third line going through our galaxy (at right angles to the two other lines), we could find the distance between two points in space with coordinates \(x^i_0 = (x^1_0, x^2_0, x^3_0)\) and \(x^i=(x^1, x^2, x^3)\), using the Pythagorean Theorem in three dimensions, to be
$$D=\sqrt{(∆𝑥^1)^2 + (∆𝑥^2)^2 + (∆𝑥^3)^2}.\tag{1}$$
Equation (1) gives us the distance \(D\) between any two points with coordinates \(x^i_0\) and \(x^i\). Since the galaxies always have fixed coordinate values, we can simply view equation (1) as the distance between any two galaxies in space. (Later on, we will come up with a “particles in the box” model where, in general, the particles will not have fixed coordinate values and it will be more useful to think of Equation (1) as the distance between coordinate points.)
Although the coordinate separation \(∆r\) between galaxies does not change, because (in general) the space can be expanding or contracting, the scaling factor \(a(t)\) (the distance \(D\) between “neighboring galaxies” whose coordinate separation is \(∆r=1\) ) will vary with time \(t\) (where \(t\) is the time measured by an ideal clock which is at rest with respect to our galaxy’s reference frame). (We shall see later on that the FRW equation determines how \(a(t)\) changes with \(t\) based on the energy density \(ρ\) at each point in space and the value of \(κ\).) Since \(a(t)\) is changing with time, it follows that the distance \(D=a(t)∆r\) between any two galaxies is also changing with time. For example, the distance \(D\) between our galaxy and other, far off galaxies is actually growing with time \(t\). The fact that the distance \(D\) between any two galaxies is changing with time according to the scaling factor \(a(t)\), this means that there must be some relative velocity \(V\) between those two galaxies as their separation distance increases with time. To find the relative velocity \(V\) between any two galaxies, we take the time rate-of-change of their separation distance \(D\) to obtain \(V=dD/dt \). \(∆r\) is just a constant and the scaling factor \(a(t)\) is some function of time; thus the derivative is
$$V=\frac{dD}{dt}=\frac{d}{dt}(a(t)∆r)=∆r\frac{d}{dt}(a(t)).$$
Let’s multiply the right-hand side of the equation by \(a(t)/a(t)\) to get
$$V=a(t)∆r\frac{d/dt(a(t))}{a(t)}.$$
\(a(t)∆r\) is just the distance \(D\) between the two galaxies moving away at a relative velocity \(V\); thus,
$$V=D\frac{d/dt(a(t))}{a(t)}.$$
The term \(\frac{d/dt(a(t))}{a(t)}\) is called Hubble’s parameter which is represented by \(H(t)\):
$$H(t)=\frac{da(t)/dt}{a(t)}.\tag{2}$$
Substituting Hubble's parameter for \(\frac{d/dt(a(t))}{a(t)}\), we get
$$V=H(t)D.\tag{3}$$
The value of Hubble’s parameter at our present time is called Hubble’s constant and is represent by \(H(today)=H_0\) . Thus, at our present time, the recessional velocities between any two galaxies is given by
$$V=H_0D.\tag{4}$$
and the value of Hubble’s constant has been measured to be
$$H_0≈500\text{ km/s/Mpc}=160\text{ km/s}.\tag{5}$$
Since \(H_0\) is a positive constant, this tells us that (at \(t=today\), not later times, because \(H(t)\) varies with time) the farther away a galaxy or object is from us (our galaxy), the faster it’s moving away. The bigger \(D\) is, the bigger \(V\) is.
By substituting Equation (5) into Equation (4) and by measuring the separation distance \(D\) between any two galaxies, we can use Equation (4) to calculate the relative, recessional speeds between those galaxies—today. To determine \(V\) as a function of time, you must compute \(a(t)\) from the FRW equation, then substitute \(a(t)\) into Equation (3); but this will be discussed later on. By substituting sufficiently big values of \(D\) (namely, values which are tens of billions of light-years) into Equation (4), one will discover that it is possible for two galaxies to recede away from one another at speeds exceeding that of light. This, however, does not violate the special theory of relativity which restricts the speeds of massive objects through space to being less than that of light. This is because it is space itself which is expanding faster than the speed of light and general relativity places no limit on how rapidly space or spacetime can expand or contract.
It might seem unintuitive, but the two coordinate points \(x^i_0\) and \(x^i\) are not actually moving through space at all. Of course, the galaxies do have some motion and velocity through space; but it is a useful idealization and approximation to assume that they are "attached" to the coordinate points and not moving through space at all. Sir Arthur Edington’s favorite analogy for this was an expanding balloon with two points drawn on its surface. As the balloon expands, the points are indeed moving away from one another; but those points are not actually moving across the space (which in this example, the space is the surface \(S^2\).)
Age of the universe
We can use Hubble’s Law to come up with a rough estimate of the age of the universe. If all of the galaxies are moving away from one another then that means that yesterday they must have been closer to one another—and a week ago even closer. If you keep running the clock back far enough, then at some time all of the galaxies and matter in the universe must have been on top of each other. Let’s assume that during that entire time interval (which we’ll call \(t_{\text{age of the universe}}\)) the recessional velocity \(V\) of every galaxy is exactly proportional to \(D\) (which, empirically, is very close to being true). Then it follows that the ratio \(D/V = 1/H\) is the same for every galaxy. Since \(1/H\) stays the same, it follows that \(1/H = 1/H_0\). Let’s also assume that during the entire history of the universe the velocity \(V\) of every galaxy remained constant. Then, according to kinematics, the time \(t_{\text{age of the universe}}\) that it took for every galaxy to go from being on top of one another (when \(D=0\)) to being where they are today is given by the equation \(t_{\text{age of the universe}}=D/V=1/H_0≈\text{14 billion years}\). (When this calculation was first performed it gave an estimate for the age of the universe of only about 1.8 billion years. Although Hubble correctly measured the recessional velocities of the galaxies, his distance measurements were off by about a factor of ten. Later astronomers corrected his distance measurements.) To come up with a more accurate age of the universe we have to account for the acceleration/deceleration of the galaxies. When we do this we are able to obtain the more accurate estimate which is given by \(t_{\text{age of the universe}}≈\text{13.8 billion years}\).
This article is licensed under a CC BY-NC-SA 4.0 license.
References
1. Leonard Susskind. "The expanding (Newtonian) universe". theoreticalminimum.com.
2. Wikipedia contributors. "Hubble's law." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 12 May. 2017. Web. 18 May. 2017.
In this lesson, we'll apply the principle of momentum conservation to analyze inelastic collisions which is when objects collide and "stick" together moving away as a single composite mass. We'll see that the conservation of momentum allows us to predict the motion after such a collision has occurred.
